Prediction of Oxygen Mass Transfer in the Presence of Plastic Carriers

Abstract

Effects of a conventional three-dimensional elastic packing (TDEP) carrier on oxygen mass transfer (OMT) efficiency in clean water were evaluated and a novel OMT model in the presence of plastic carriers in wastewater treatment was proposed. Results showed that OMT efficiency was increased in the presence of TDEP carriers, which had three main positive effects: (1) shearing bubbles; (2) changing bubble moving tracks; (3) enhancing bubble dispersion, and one passive effect: reducing the velocity magnitude and turbulence kinetic energy of mixed liquid. A novel OMT model was established to provide a new OMT coefficient expression. Compared with the previously established model, this novel OMT model represents actual wastewater treatment operation more closely, and provides a better knowledge for OMT efficiency in the presence of the carriers in wastewater treatment.

Introduction

Biofilm processes such as moving bed biofilm reactors (MBBR), integrated fixed-film activated sludge (IFAS), and biological aerated filters, play a significant role in treating domestic and industrial wastewater because of their high pollutant removal efficiency. The effects of the carrier used in oxygen mass transfer (OMT) have been evaluated in the following studies. A study on the effects of plastic support carrier on the diffusion of air in a submerged aerated filter found that the performance of a coarse-bubble diffuser was enhanced by the addition of carriers as indicated by a higher oxygen transfer efficiency (OTE) and oxygen transfer rate (OTR). Further, they demonstrated that the performance of a fine-bubble diffuser was inhibited by the carrier as indicated by falling OTR and OTE values (Hodkinson et al., 1998). Many studies on the MBBR process demonstrated that a suspended carrier enhanced the oxygen transfer ability, and this enhancement proportionally increased with the filling rate when the carrier was allowed to flow freely and uniformly (Cao and Zhang, 2003; Wang et al., 2005; Jie et al., 2009). Some researchers reported the effect of plastic carrier media in IFAS via the off-gas method, and found that the carrier had little or no effect on OTEs for IFAS. Further, they showed that the average process water standard OTEs of experiments in the presence and absence of carrier were comparable (Xu et al., 2000; Viswanathan et al., 2008).

Generally, the OMT performances in the biofilm processes are measured in clean water, before the starting up of a wastewater treatment plant, according to a standardized procedure (ASCE, 1992; NFEN, 2004). Four parameters including the apparent volumetric mass transfer coefficient (K_La), the standard oxygen transfer rate (SOTR), the standard oxygen transfer efficiency (SOTE), and the standard aeration efficiency (SAE) are commonly used for the assessment of OMT performance.

A number of mass transfer theories, such as the two-film theory proposed by Lewis and Whitman (1924), Higbie's penetration theory, and Danckwerts' (1951) surface-renewal theory, have been proposed to explain the mechanism of oxygen transfer from air to liquid. Considering the effects different parameters have on OMT, the above theories cannot accurately explain the OMT characteristics in a biofilm reactor because of the complexity of OMT in the presence of carrier. Thus, the development of a suitable model is necessary for assessing the role of carrier in the biofilm-based wastewater treatment process. The objective of this work is to investigate the influence of TDEP carrier, which is mainly used in the wastewater treatment process (Li et al., 2005), on K_La₂₀, SOTR, SOTE, and SAE in clean water under fine-bubble aeration conditions and to determine the underlying mechanisms.

Experimental Materials and Methods

Materials

The industrial grade Na₂SO₃ as oxygen elimination agent and industrial grade COCl₂·6H₂O as catalyst were used. Tap water with a total dissolved solids concentration of 300 mg/L was used as the test water. For the carrier, TDEP is made of polyolefin and has a density of 0.93 g/cm³ (Yixing Renyuan Environmental Protection Co., Ltd.), which is shown in Fig. 1. The surface area per unit volume is 200 m²/m³ and the void fraction (ɛ) is greater than 0.99. The length of every yarn is about 3 m, and the yarn diameter is 150 mm.

FIG. 1.

TDEP carriers. TDEP, three-dimensional elastic packing.

Apparatus

The pilot experiment setup is shown in Fig. 2, which consisted of a plexiglass column (1.5 m in internal diameter ×7 m in height) with the effective volume of 10.60 m³. A corundum porous fine-bubble diffuser with a diameter of 245 mm was placed at the bottom of the column. Other components included an air blower, a charge-coupled device (CCD) high-speed videography system, a calibrated rotameter, three dissolved oxygen (DO) probes placed at three representative points, and an automatic recorder. The air flow rate was designed to range from 2.0 to 5.0 m³/h, while the TDEP carriers packing density (CPD) varied from 0% to 30%. TDEP carriers were tied to the strings that were set on the aeration tank surface side by side and hung with weighted materials at the end of yarn.

FIG. 2.

Schematic diagram for the fine bubble aeration system: (a) data logger; (b) D/A converter; (c) pressure gauge; (d) DO meter; (e) DO probe; (f) rotameter; (g) aeration tank; (h) ceramic diffuser; (i) camera; (j) video tape recorder; (k) screen; (l) TDEP. DO, dissolved oxygen.

Methods

Experimental procedure

The unsteady-state clean water test consisting of deoxygenation and reoxygenation periods was conducted in the aeration tank. Static deoxidization was obtained by adding a deoxidizer (sodium sulfite) and a catalyst (cobalt chloride hydrate). DO was measured by DO4200 (Shsanbena) with the range from 0 to 20 mg/L. When the DO values reached zero, aeration started until the DO concentration in the water achieved saturation. During the reoxygenation period, the DO values were monitored online using membrane probes in situ at several representative points in the aeration tank (CJ/T 3015.2, 1993; ASCE, 2007).

Parameter determination

The K_La based on water temperature and the DO concentration was estimated by using nonlinear regression. The water temperature was between 20°C and 25°C during the test. The K_La₂₀, SOTR, SOTE, and SAE can be calculated according to the ASCE STANDARD (ASCE, 2007) methods. Abbreviation of the terms is listed in Table 1.

Table 1.

Abbreviation of Terms

Abbreviations	Terms
BAF	Biological aerated filter
CCD	Charge-coupled device
CFD	Computational fluid dynamics
CPD	Carrier packing density
IFAS	Integrated fixed-film activated sludge
K_La	Apparent volumetric mass transfer coefficient
MBBR	Moving bed biofilm reactors
OMT	Oxygen mass transfer
OTE	Oxygen transfer efficiency
OTR	Oxygen transfer rate
SOTE	Standard oxygen transfer efficiencies
SOTR	Standard oxygen transfer rate
SAE	Standard aeration efficiency
TDEP	Three-dimensional elastic packing
TDS	Total dissolved solids

Results and Discussion

Effects of TDEP on OMT in the fine-bubble aeration system

Table 2 shows the experimental results of K_La₂₀, SOTR, SOTE, and SAE in the presence and absence of TDEP, indicating that the OMT was improved by the addition of TDEP carrier in fine-bubble aeration system. Compared with no carrier addition, the K_La₂₀, SOTR, SAE, and SOTE were all found to considerably increase at various air flow rates (2, 3, 4, and 5 m³/h) when the CPD was 23.9%. For example, the K_La₂₀, SOTR, SOTE, and SAE increased by 36.0%, 36.0%, 36.5%, and 36.4%, respectively, at an air flow rate of 2 m³/h.

Table 2.

Experimental Results Shown in the Presence and Absence of TDEP

	K_La₂₀ (min⁻¹)		SOTR (kg/h)		SOTE (%)		SAE (kg/[kW·h])
Air flow rate (m³/h)	1 ^a	2 ^b	1 ^a	2 ^b	1 ^a	2 ^b	1 ^a	2 ^b
2	0.025	0.039	0.145	0.227	19.8	31.1	4.44	6.98
3	0.037	0.050	0.212	0.293	19.3	26.7	4.33	5.99
4	0.047	0.059	0.271	0.342	18.4	23.6	4.14	5.28
5	0.050	0.071	0.290	0.410	15.8	22.6	3.55	5.05

The experimental results in the absence of TDEP.

The experimental results in the presence of TDEP (CSR: 23.9%).

TDEP, three-dimensional elastic packing; K_La, apparent volumetric mass transfer coefficient; SOTR, standard oxygen transfer rate; SOTE, standard oxygen transfer efficiency.

The influence of the CPD on OMT was also investigated at a gas flow rate of 3 m³/h (Fig. 3). We found that the K_La₂₀, SOTR, SOTE, and SAE increased rapidly when the CPD increased from 0% to 17.9%, which remained almost stable when the CPD was over 17.9%. This can be explained by the fact that only some of the bubbles contacted the TDEP at low CPD (<17.9%), leading to a notable effect on OMT. When the CPD increased continuously over 17.9%, all bubbles interacted with the TDEP leading to a decrease of the effect on OMT. These results are similar to some studies that found OMT could be enhanced in the presence of TDEP carriers (Huang et al., 1992). However, their studies did not address the effects of the CPD on OMT. This discrepancy can be attributed to the differences in experimental conditions and the aeration system. In addition, during the MBBR processes, many researchers found that the presence of the carriers enhanced OMT, which was improved with the increasing of CPD when the carriers were allowed to move freely driving by the water current, while decreased slightly when the movement of the suspended carrier was restricted (Yang et al., 2000; Cao and Zhang, 2003; Wang et al., 2005; Jie et al., 2009).

FIG. 3.

Influence of CPD at an air flow rate of 3 m³/h. CPD, carrier packing density.

Influencing mechanism of TDEP on OMT

Effects of TDEP on gas phase in the fine-bubble aeration system

To better understand the role of the carrier on the gas phase, bubble images were taken by CCD camera in a laboratory-scale tank (0.4 m in length ×0.4 m in width ×0.65 m in height) at gas flow rates of 0.1–0.5 m³/h and a CPD of 0–29.4%. The CCD images show that TDEP had three main influences on the gas phase: bubble shearing (Fig. 4), changing the bubble movement tracks and enhancing bubble dispersion. As shown in Fig. 5, the size of the bubble just before leaving the water was smaller in the presence of TDEP than in the absence of TDEP. For example, at an air flow rate of 0.2 m³/h and a CPD of 5.9%, TDEP reduced the size of the final bubbles form 3.9 to 3.7 mm. TDEP reduced the size of the final bubbles form 4.3 to 3.6 mm at an air flow rate of 0.5 m³/h and a CPD of 5.9%. Thus, the surface area per unit volume (a) of the bubble was increased due to the decrease of bubble size, indicating that the shearing could enhance the OMT. It can also be found that the effect of TDEP on bubble size was more significant for the higher air flow rate than the lower. The final bubble size was affected by the addition of carrier regardless of the air flow rate, which was comparable at all air flow rates in the presence of the TDEP carrier with a consistent CPD, while the final bubble size increased with increasing air flow rates in the absence of the TDEP.

FIG. 4.

Bubble shearing in the presence of TDEP: the encircled bubbles demonstrate an example of bubble shearing. (1) shows that a bubble comes close to a fiber; (2) shows that a bubble is going to be cut down. (3) and (4) show that one bubble had been cut into two small bubbles.

FIG. 5.

Initial bubble just leaving from diffuser and the final bubble size just before leaving aeration tank.

The bubble moving images were captured by the CCD camera. Bubble movement tracks included lateral movements along the carrier wire drawings, stationary movements, vertical movements, and rebound movements in the presence of TDEP carriers. These complex movement tracks can extend the retention time of the bubbles in the liquid, and thus increase the OMT flux.

To further understand the effects of TDEP on the bubble movement tracks, we calculated the probability of the various bubble movement tracks at a CPD of 5.9% and at different gas flow rates. The lateral movements were found to be the most prevalent of all movement tracks in the presence of TDEP. Although increasing the gas flow rate increased the probability of this type of movement, it had a tendency to decrease stationary and vertical movements, and had little effect on rebound movements.

Effect of TDEP on liquid phase in the fine-bubble aeration system

To better understand the role of the carrier on the liquid phase, a computational fluid dynamics procedure utilizing the Eulerian–Eulerian approach has been used to solve the governing differential equations for the gas–solid–liquid mass transport problem based on the standard k − ɛ model incorporating additional terms that take account of the interfacial turbulent momentum transfer.

Figure 6(A, B) shows the simulation results of velocity magnitude of mixture at gas flow rates of 0.2 m³/h with a CPD of 0% and 5.9%. The results indicated that: (1) TDEP carriers reduced velocity magnitude of mixture, the discrepancy was small; (2) the continuously varying trend of axial and longitudinal velocity in the presence of TDEP was in accordance with that in the absence of TDEP, although the former fluctuated greatly.

FIG. 6.

Velocity magnitude of mixture: (A) axial velocity magnitude, (B) longitudinal velocity magnitude.

Figure 7(A, B) is the simulation of turbulence kinetic energy of mixture in the presence and absence of TDEP carriers. The results indicated that the turbulence was reduced by TDEP carriers not only in the turbulence kinetic energy but also in the region of turbulence, while the development tendency of both turbulences was similar.

FIG. 7.

Turbulence kinetic energy of mixture: (A) axial turbulence kinetic energy, (B) longitudinal turbulence kinetic energy.

It can be seen from the results that plastic TDEP carriers had complicated effects on OMT. The positive effects included reduced bubble size and extended retention time of the bubbles in the liquid phase. The passive effects included reduced velocity magnitude and turbulence kinetic energy of mixture. It could also be inferred from the results that the positive effects had greater effect on OMT so that OMT could be enhanced by TDEP carriers in clean water.

Establishment of mass transfer model

From the results above, the carrier can affect the OMT because it can change the bubble movement track and thus extend the retention time of the bubbles in the liquid phase. Therefore, in the presence of carrier, the establishment of the OMT model should consider the effects of the carrier on the changes to the bubble movement tracks.

(1) The hypothesis for the establishment of a new model. Based on our findings, a novel model was established to more accurately assess OMT in the presence of carrier. Postulates for this model are as follows: (i) The diffusion between gas and water at the gas–liquid interface is nonstationary; (ii) the depth of the bubbles in the water is shallow; (iii) the resistance of liquid film on mass transfer is considered, while the resistance of gas film is ignored (Xu and Long, 1991); (iv) the effects of bubble curvature are ignored, and thus the mass transfer area is equal to the surface area of the bubbles (Xu and Long, 1991; Sheng and Qiao, 2000); (v) the liquid velocity gradient produced by rising bubbles is ignored, and the contact time gradient between the gas and liquid caused by different types of movement is also ignored; (vi) the probability of lateral movement, stationary movement, vertical movement, and rebound movement probabilities are A(%), B(%), M(%), and N(%), respectively, where A+B+M+N=100(%).

(2) OMT modeling. Bubble movements along a certain path with a certain speed (v) can be viewed as movement of the fluid element around the bubble in the opposite direction with the same speed (v). The fluid element outside the bubble can be used for analyzing the balance of bubble OMT in water. Figure 8 shows a schematic diagram of a fluid element responding to the lateral movement of a bubble. dydz represents the fluid element, the vertical direction is marked as the z axis, the horizontal direction as the y axis, the mass flux generated by the convective motion of the water in the direction of the z axis as N_0,z, and the mass diffusion flux generated by the concentration difference of oxygen as N_0,y (Xu and Long, 1991).

FIG. 8.

Schematic diagram representing the fluid element outside of a bubble.

According to the mass conservation law of the infinitesimal fluid element (dydz), the following formula can be obtained (Xu and Long, 1991): \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}&{N_{o , y} dz + N_{o , z} dy + dydzr_0} \\& \quad= {N_{o,y + \Delta y} dz + N_{o,z + \Delta z}dy + (\partial c / \partial t) dydz} \tag{1}\end{align*} \end{document}

r₀ is the oxygen uptake rate of microorganisms. Because of the absence of microorganisms in clean water, r₀=0. Equation (1) can be reformulated to the following equation: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} &- D \bigg ( {\partial c \over \partial y} \bigg) dz + vc \cos \theta dz + vc \sin \theta dy \\& \quad\quad= - D{ \partial \over \partial y} \bigg( c + dy {\partial c \over \partial y} \bigg) dz + v \cos \theta \bigg( c + dy {\partial c \over \partial y}\bigg) dz \\& \quad\quad\quad + v \sin \theta \bigg( c + dz {\partial c \over \partial z}\bigg) dy + { \partial c \over \partial t} dydz \tag{2}\end{align*} \end{document}

Equation (2) can be changed: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}v \sin \theta ( \partial c \big/ \partial z ) + v \cos \theta ( \partial c \big/ \partial y ) - D ( \partial^2 c \big/ \partial y^2 ) + \partial c \big/ \partial z = 0 \tag{3}\end{align*} \end{document}

Because of the relationship \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\partial z = v \sin \theta \partial t ;$$ \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\partial y = v \cos \theta \partial t$$ \end{document} , Equation (3) could be simplified: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}\partial c \big/ \partial z = D \cdot \partial^2 c \big/ 3 \partial y^2 \tag{4}\end{align*} \end{document}

Boundary conditions of differential equations:

(1) when t=0 and y>0, c=c_b;

(2) when t>0 and y=0, c=c* and;

(3) when t>0 and y=∞ , c=c_b.

According to the Laplace transformation solution, Equation (5) can be obtained: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}\partial c \big/ \partial y \mid_{y = 0} = - \sqrt{3} \cdot { ( c^* - c_b ) } \big/ \sqrt{ \pi t D} \tag{5}\end{align*} \end{document}

The gas mass flux is given: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}N_0 = - D ( \partial c \big/ \partial y ) _ {y = 0} = \sqrt{3D} \cdot { ( c^* - c_b ) } \big/ \sqrt{ \pi t_c} \tag{6}\end{align*} \end{document}

Because the penetration theory holds that OMT is transient, the average flux within a short time (t_c) is calculated: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \overline{N}_0 =& {1 \over t_c} \int ^{t_{c}}_0 \sqrt{3D} \cdot \left[ ( c^* - c_b ) \big/ \sqrt{ \pi t} \right] dt \\ =& 2 \sqrt{3D} \cdot ( c^* - c_b ) \big/ \sqrt{ \pi t_c} \tag{7}\end{align*} \end{document}

Thus, the bubble mass flux based on the lateral movement is given: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}\overline {N}_{01} = 2A \sqrt{3D} \cdot ( c^* - c_b ) \big/ \sqrt{ \pi t_c} \tag{8}\end{align*} \end{document}

For the other types of bubble movements including stationary, vertical, and rebound, the establishment of material balance and boundary conditions were similar to lateral movements. Therefore, the bubble mass flux based on the stationary, vertical, and rebound movements can be given, respectively, as follows: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}\overline {N}_{02} = 2B \sqrt{D} \cdot ( c^* - c_b ) \big/ \sqrt{ \pi t_c} \tag{9}\end{align*} \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}\overline {N}_{03} = 2M \sqrt{2D} \cdot ( c^* - c_b ) \big/ \sqrt{ \pi t_c} \tag{10}\end{align*} \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}\overline {N}_{04} = 2N \sqrt{2D} \cdot ( c^* - c_b ) \big/ \sqrt{ \pi t_c} \tag{11}\end{align*} \end{document}

For the simultaneous production of these bubble movement tracks, a new OMT model within a short time (t_c) in the presence of the TDEP can be obtained: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \overline {N}_0 =& \overline {N}_{01} + \overline {N}_{02} + \overline {N}_{03} + \overline {N}_{04} \\=& \big( 2 \sqrt{D} \big/ \sqrt{ \pi t_c} \big) \big( \sqrt{3} A + B + \sqrt{2} M + \sqrt{2}N \big) (c^* - c_b ) \tag{12}\end{align*} \end{document}

Therefore, a new OMT coefficient can be written as follows: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}K_L = \big( 2 \sqrt{D} \big/ \sqrt{ \pi t_c} )\big ( \sqrt{3}A + B + \sqrt{2}M + \sqrt{2}N ) \tag{13}\end{align*} \end{document}

Taking into account the bubble size in Equation (18), the following equation is given: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}K_L = \big(2 \sqrt{Dv} \big/ \sqrt{ \pi d} )\big ( \sqrt{3} A + B + \sqrt{2}M + \sqrt{2}N ) \tag{14}\end{align*} \end{document}

Where K_L is proportional to D^0.5, which is in agreement with the penetration theory and the surface renewal theory in the new model, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$(\sqrt{3} A + B + \sqrt{2} M + \sqrt{2} N )$$ \end{document} >1, K_L in the new model is greater than that in penetration theory, where K_L is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$2 \sqrt{D} \big/ \sqrt{ \pi t_c}$$ \end{document} . This is because the influence of the various bubble tracks that had great positive enhanced effects on OMT was taken into consideration in this new model, and this is the basic reason of discrepancy in those two models.

For the two-film theory, the K_L is proportional to D based on the hypotheses of two layer static membranes at the gas–liquid phase interface and ignoring the effect of rising bubble movement on OMT. For the surface-renewal theory, fluid element exposure time is introduced, which can be an arbitrary value between 0 and ∞. But it ignores one crucial fact: the update of the fluid element on the gas–liquid interface is generated by bubble movement instead of liquid turbulence (Sheng and Qiao, 2000). The update of the fluid element on the gas–liquid interface is generated by bubble movement instead of liquid turbulence, which was taken into account in the penetration theory that only considered the bubble upward movement. In fact, in the presence of TDEP, movement tracks are complex and the lateral movements were found to be the most prevalent of all movements, which is ignored by the penetration theory, so it is not very suitable for the actual situation. However, these hypotheses are not suitable for the actual situation in the presence of TDEP, so the new model was a development of conventional mass transfer theories.

Other equations used for estimating K_L have also been reported in previous studies (Xu and Long, 1991; Sheng and Qiao, 2000);. For example, studies (Sheng and Qiao, 2000) have reported that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$K_L = \sqrt{2DV / \pi d}$$ \end{document} , taking into account only the upward movement of the bubbles. The results showed that (i) when Re=1, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$K_L = 0.65 \sqrt{DV / d}$$ \end{document} ; (ii) when Re>1,000, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$K_L = 1.13 \sqrt{DV / d}$$ \end{document} , which were deduced from the Sherwood number expression and without considering the specific conditions in the presence of TDEP.

Therefore, the new OMT model and the new OMT coefficient expression proposed in this study are much closer to the actual wastewater treatment operation situation, and thus provide a better knowledge for OMT efficiency in the presence of the TDEP carrier in wastewater treatment. Because different kinds of carriers may have different effects on OMT, the OMT model of this study for the application in other carriers needs to be further studied in future.

Conclusions

Effects of TDEP carrier on OMT were investigated and the following conclusions can be indicated from this study. OMT could be enhanced by TDEP carrier, which is indicated by the higher values of K_La₂₀, SOTR, SOTE, and SAE. The mechanism of TDEP in enhancing OMT is to shear bubbles, to change the bubble movement tracks, and to enhance the bubble dispersion, which could reduce the size of bubbles, and to extend the retention time of the bubbles in the liquid phase. Meanwhile, the velocity magnitude and turbulence kinetic energy of mixture could be reduced by the presence of TDEP. The new OMT model established in this study involves the main bubble movements in the presence of the TDEP and therefore more closely represents the actual operation for aeration tanks in wastewater treatment.

Footnotes

Acknowledgments

This work was part of the Project Effect of Suspended Carriers on Oxygen Mass Transfer Process and Its Mechanism in Aeration of Moving Bed Biofilm Reactor supported by the National Natural Science Foundation of China (No. 51408601).

Author Disclosure Statement

No competing financial interests exist.

\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\overline{\rm N}_{04}$$ \end{document} =the average flux generated by the rebound movements of the bubbles

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